Draft:−3

• Comment: Per comment by User:Hurricane Wind and Fire, needs a properties section on a par with -2. Stuartyeates (talk) 23:15, 13 June 2026 (UTC)

• Comment: Still largely just a dictionary definition. Expand onto the properties section to a point where it is similar to the properties section in -2 and it will be ready for acceptance. 🌀Hurricane Wind and Fire, why did you decline my draft? (talk) (contribs)🔥 03:41, 12 June 2026 (UTC)

• Comment: If you want the draft to be like -2, then please include citations so this is not a dictionary entry GGOTCC 21:07, 1 November 2025 (UTC)

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← −4 −3 −2 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers
Cardinal	negative three
Ordinal	-3rd (negative third)
Divisors	1, 3
Arabic	−٣
Bengali	−৩
Binary (byte)	11111101

In mathematics, −3 (negative three or minus three) is an negative integer three units from the origin, denoted as −3 or ⁻3. It is the additive inverse of 3, positioned between −4 and −2. It is the second largest negative odd number.

Properties

• Negative three is a quadratic residue modulo 4:

$1^{2}\equiv -3{\pmod {4}}$

• Negative three is the first, and smallest negated heegner number. only nine negative numbers have this property: −3, −4, −7, −8, −11, −19, −43, −67, −163.
• Negative three is the first and last weight of Spencer's 15-Point Moving Average.

Divisors of negative three

The divisors of the number negative three, including negative divisors, are identical to those of two: 1, 3, −1 −3. Since its only divisors are ±1 and ±3, negative three is considered an irreducible element, which is the equivalent of a prime number for negative integers.

Square root of negative three

The square root of negative three produces both quadratic field, and cyclotomic field.

In the quadratic field $\mathbb {Q} ({\sqrt {-3}})$, negative three is its fundamental discriminant, and the ring of integers in this field is the lattice of Eisenstein integers.

${\mathit {Z}}[\omega ],\omega ={\frac {-1+{\sqrt {-3}}}{2}}$

which forms a perfect hexagonal (triangular) lattice in the plane.

The quadratic field also has class number 1, meaning it behaves better as the integers for factoring.

In the cyclotomic field $\mathbb {Q} (\zeta _{3})$, the discriminant of the field is −3, meaning three is the only rational prime that becomes ramified, and the ramification index is exactly two.

Representation

Negative three is typically represented with a negative sign before 3.

In a balanced ternary numeral system, negative three is represented perfectly by a single digit shifted by a place value. While positive three is written as 10, negative three is represented as T0 (or 1̄0), where "T" represents −1.

See Also

• 3

References

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Categories:

• Integers